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The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry (Paperback), 20 Nov 2011

I found this book to be a great introduction to group theory from a lay perspective. I had tried to teach myself group theory from textbooks and even from Penrose's Road to Reality (differs little from a textbook) but struggled. The definitions are abstract and somewhat circular; self-consistent but too abstruse. Funnily enough this book starts from the very beginning and explains what a group is and how it relates to symmetry. All of a sudden I understood. Telling me what a group is with a few real life examples really hit the idea home. That groups relate to symmetry make it very easy to figure out what their use is.

Livio starts from the very basics and assumes very little, no real mathematical knowledge is needed. It definitely isn't a group theory textbook for the layperson, and having been able to flick through the book before purchase I definitely knew there wasn't many equations. The title of the book had me wondering which equation Livio actually meant; however, the subtitle really gives it away: 'language of symmetry'. If you have read Simon Singh's "Fermat's Last Theorem" then you'd probably already know of the failure to find a general equation that gives the roots of all quintic polynomials (the titular equation of Livio's book). I found this book to be far more readable and quicker to finish than Singh's book.

Livio follows the history of group theory and shows how it eventually becomes the language of symmetry. The book eventually covers the life of Lie and Galois and their roles in the discovery of (1) the relationship between the roots of quintic polynomials and (2) group theory. For what it's worth, there is a way of generating the roots of quintic polynomials but not using the normal methods (i.e. via radicals) as we would for say the quadratic equation, this is what Galois essentially proved.

The book concludes with why Livio believes symmetry to be important, not only in a pure mathematical sense but how it applies to society at large. Symmetry is a ubiquitous theme in nature, it plays a role in evolution (mate selection) as well as our musical tastes (which in turn is likely dependant upon evolution).